Lemma 22.1.97E61430B531F3

Lemma 22.1.97.E61430B531F3 Let $L$ be a big line bundle on $X$ such that $c_1(L) = \{ \theta \} $ and $\Phi \in \mathrm{PSH}(X\times \Delta , p_1^*\theta )$, where $\Delta $ is the unit disk. Let $G$ be an ample line bundle on $X$. Then there exists $k>0$, depending only on $X$ and $G$, such that $p_1^*(G^ k \otimes L^ m)\otimes \mathcal I(m \Phi )$ is globally generated for all $m \in \mathbb {Z}_{>0}$.